Taylor expansion/R/One variable/Introduction/Section

So far, we have only considered power series of the form ${}\sum _{k=0}^{\infty }c_{k}x^{k}$. Now we allow that the variable ${}x$ may be replaced by a "shifted variable“ ${}x-a$, in order to study the local behavior in the expansion point ${}a$. Convergence means, in this case, that some ${}\epsilon >0$ exists, such that for

Brook Taylor (1685-1731)

${}x\in ]a-\epsilon ,a+\epsilon [\,,$

the series converges. In this situation, the function, presented by the power series, is again differentiable, and its derivative is given as in fact. For a convergent power series

${}f(x):=\sum _{k=0}^{\infty }c_{k}(x-a)^{k}\,,$

the polynomials ${}\sum _{k=0}^{n}c_{k}(x-a)^{k}$ yield polynomial approximations for the function ${}f$ in the point ${}a$. Moreover, the function ${}f$ is arbitrarily often differentiable in ${}a$, and the higher derivatives in the point ${}a$ can be read of from the power series directly, namely

${}f^{(n)}(a)=n!c_{n}\,.$

We consider now the question whether we can find, starting with a differentiable function of sufficiently high order, approximating polynomials (or a power series). This is the content of the Taylor expansion.

Definition

Let ${}I\subseteq \mathbb {R}$ denote an interval,

$f\colon I\longrightarrow \mathbb {R}$

an ${}n$-times differentiable function, and ${}a\in I$. Then

${}T_{a,n}(f)(x):=\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}\,$

is called the Taylor polynomial of degree ${}n$ for ${}f$ in the point ${}a$.

So

${}T_{a,0}(f)(x):=f(a)\,$

is the constant approximation,

${}T_{a,1}(f)(x):=f(a)+f'(a)(x-a)\,$

is the linear approximation,

${}T_{a,2}(f)(x):=f(a)+f'(a)(x-a)+{\frac {f^{\prime \prime }(a)}{2}}(x-a)^{2}\,$

is the quadratic approximation,

${}T_{a,3}(f)(x):=f(a)+f'(a)(x-a)+{\frac {f^{\prime \prime }(a)}{2}}(x-a)^{2}+{\frac {f^{\prime \prime \prime }(a)}{6}}(x-a)^{3}\,$

is the approximation of degree ${}3$, etc. The Taylor polynomial of degree ${}n$ is the (uniquely determined) polynomial of degree ${}\leq n$ with the property that its derivatives and the derivatives of ${}f$ at ${}a$ coincide up to order ${}n$.

Theorem

Let ${}I$ denote a real interval,

$f\colon I\longrightarrow \mathbb {R}$

an ${}(n+1)$-times differentiable function, and ${}a\in I$ an inner point of the interval. Then for every point ${}x\in I$, there exists some ${}c\in I$ such that

$f(x)=\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}+{\frac {f^{(n+1)}(c)}{(n+1)!}}(x-a)^{n+1}.$

Here, ${}c$ may be chosen between

${}a$ and ${}x$.

Proof

This proof was not presented in the lecture.

$\Box$

The real sine function, together with several approximating Taylor polynomials (of odd degree).

Corollary

Suppose that ${}I$ is a bounded closed interval,

$f\colon I\longrightarrow \mathbb {R}$

is an ${}(n+1)$-times continuously differentiable function, ${}a\in I$ an inner point, and ${}B:={\max {\left(\vert {f^{(n+1)}(c)}\vert ,c\in I\right)}}$. Then, between ${}f(x)$ and the ${}n$-th Taylor polynomial, we have the estimate

${}\vert {f(x)-\sum _{k=0}^{n}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}}\vert \leq {\frac {B}{(n+1)!}}\vert {x-a}\vert ^{n+1}\,.$

Proof

The number ${}B$ exists due to fact, since the ${}(n+1)$-th derivative ${}f^{(n+1)}$ is continuous on the compact interval ${}I$. The statement follows, therefore, directly from fact.

$\Box$